The ezpected ambiguity function (EAF) is shown to provide a generalization of stationary correlation analysis to nonstationary random processes. Important properties of the EAF are discussed, and the EAFs of special processes are considered. Based on the EAF, a fundamental classification (underspread/overspread) of nonstationary processes is introduced and shown to be relevant to timevarying spectral analysis.
I N T R O D U C T I O NThe correlative analysis of stationar processes using theis of fundamental importance [l]. In particular, the power spectrum S,(f) is the Fourier transform of the ACF ( Wiener-Khintchine relation)',For a nonstationary process x ( t ) , the ACF r z ( t i , t z ) = E { x ( t i ) x*(t2)} is a 2-D function [l], and the power spectrum S,(f) is replaced by a time-varying power spectrum T,(t, f ) such as the physical spectrum, the (generalized) Wagner-Ville spect m m , or the evolutionary spectrum [2, 3, 41. Nonstationary processes exhibit spectral correlation [5] as measured by the spectral ACF R Z ( f 1 , f 2 )= E{X(f')X*(fi)} (assuming existence of the Fourier transform X ( f ) of z(t)).We now ask if there exists a joint time-frequency (TF) correlation function which combines the temporal ACF r , ( t i , t 2 ) and the spectral ACF R , ( f l , f i in a meaningful way, and which is related to a meanin d ul time-varying spectrum by a Fourier transform (generalization of the Wiener-Khintchine relation (1)). In this paper, we show that a satisfactory answer to this question is provided by the expected ambiguity function (EAF) recently proposed in [S, 7 . We demonstrate that the effective support region nonstationarity inherent in a rocess. We then introduce a fundamental classification &nderspread/overspread) of nonstationary processes. For underspread processes, various timevarying spectra (such as the generdized WignerVille spectra and the evolutionary spectra) are shown to be effectively equivalent. Furthermore, the underspread property is relevant to timevarying spectrum estimation, and finally, the physical spectrum of an underspread process is a complete second-order description of the process. 'Funding by grant 4913 of the Jubilaumsfondf der Osterrei-'Integrals go from -cu to 00 unless specified otherwise. of the k A F provides useful indications about the type of chischen Nationalbank and FWF grant The (generalized) ambiguity function (AF) [SI of a signal x ( t ) is where a is a realvalued parameter. We define the ezpected (generalized) ambiguity function (EAF) of a nonstationary random process s(t) as the expectation of the AF, EA?)(^, e' E{A?)(~, VI} .It follows that the EAF is the Fourier transform of the a-parameterized ACF with respect to t ,
J tSince the ACF can be recovered from the EAF by inversion of the Fourier transform (2) followed by a simple substitution to obtain r , ( t l , t z ) from r p ) ( t , T ) , the EAF is a complete second-order description of the process.Interpretation as TF Correlation. An intuitively reasonable measure for the statistical correlation between ...
